One great 'tmnt and batman fanfiction' is 'The Dark Turtles'. It combines the grittiness of Batman's Gotham with the fun and unique personalities of the Teenage Mutant Ninja Turtles. The story has an interesting plot where the Turtles end up in Gotham and have to team up with Batman to fight a common enemy.
One possible storyline could be a battle of wits between Leonardo and Tigerclaw. Leonardo, being the leader of the Turtles, would constantly try to outsmart Tigerclaw. For example, he could set up elaborate traps to catch Tigerclaw and stop his evil plans. Tigerclaw, on the other hand, would be constantly trying to anticipate Leonardo's moves and turn them against him. Another storyline could be a more complex relationship where they start to understand each other's motives over time. Maybe there could be a situation where they have to team up against a common, even greater enemy. This could lead to a begrudging respect between them.
One idea could be a story where Leonardo and Tigerclaw are forced to team up temporarily to defeat a common enemy greater than their rivalry. Maybe a new, powerful mutant emerges that threatens both the turtles and Shredder's plans. This would put them in a situation where they have to rely on each other's skills.
The conflict between good and evil is a key element. Leonardo represents the good, and Tigerclaw is clearly on the side of evil. Their clashes, both physical and ideological, can create a lot of tension and excitement. For instance, their different fighting styles - Leonardo's ninjutsu training with Splinter versus Tigerclaw's more brute - force and underhanded tactics. Another element is the mystery around Tigerclaw. There could be parts of his past that are slowly revealed in the fanfiction, which could add depth to the story.
DeviantArt can also be a source for this kind of fanfiction. While it's mainly known for art, there are some writers who post their stories there as well. You can search using relevant tags like 'tmnt batman crossover fanfiction' to narrow down your search.
One of the special things about this graphic novel could be the character development. Since the two groups of characters are so different, they will likely have to adapt and grow as they interact. Batman is a brooding, dark hero, while the TMNT are more light - hearted and youthful. Their interactions could lead to some really interesting character arcs, and the graphic novel may also have some great visual representations of Gotham City and the TMNT's sewer lair.
Obviously, Batman is one of the main characters. He's the Dark Knight from Gotham City. And then there are the Teenage Mutant Ninja Turtles - Leonardo, Donatello, Michelangelo, and Raphael. These are the core characters that are likely to drive the story forward in this graphic novel.